This invention relates to a method and apparatus for determining the frequency content of a time series.
A digital signal comprises a series of discrete valued elements (or samples) known as a time series. It is often desired to determine the frequency content of a digital signal to, for example, allow detection of a tone in a noisy signal. A frequency content determination is necessarily based on a segment of the signal over a limited time interval. This signal segment may be obtained by multiplying the time series by a function which has a time interval equal to the time interval chosen for the segment. The function is known as a "window" and the simplest window is a rectangular window which truncates the time series to the chosen time interval without altering the relative values of elements of the segment. A window is also represented by a series of discrete valued (quantised) elements. A discrete Fourier transform of the product provides information on the frequency content of the segment.
One drawback of applying a window to a time series is that the window results in frequency "leakage". That is, any given frequency in a time series will, after multiplication with a window, spread out over the whole frequency range. The result in the Fourier domain is sidelobes around each lobe representing a frequency in the series. With a rectangular window, these side lobes are large and may interfere with proper identification of the main lobes which represent the frequencies actually in the time series.
Another problem with a rectangular window is that the discontinuities of the window result in oscillations in the frequency domain no matter how many discrete elements make up the window. This is known as the Gibbs phenomenon.
To address the problem of leakage as well as the Gibbs effect, a number of non-discontinuous non-rectangular windows have been developed. Such windows are symmetric and taper to zero or to a smaller non-zero value at either end. An exemplary non-rectangular window is the Blackman window, which is a triangular window. The weighting that these windows apply to the time series segment reduces the size of the sidelobes. However, these tapered windows also increase the width of the main lobes, which decreases frequency resolution (i.e., two closely spacely frequencies may be seen as one lobe). The resolution problem may be combatted by increasing the number of samples in the segment (i.e., increasing segment time interval), however, this increases processing time, and so there is a trade-off to address in any design.
A digital fixed point representation of a non-rectangular window is generally an approximation of the true window shape with the degree of approximation depending upon how many levels of quantisation are available to represent each point in the window. I have recognised that, because of this, a digitally represented window may have zero-valued end portions which reduce the number of elements of the time series segment contributing to the frequency determination.
This invention seeks to overcome drawbacks of known windowing systems.